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An algebra whose additive group can be represented in the form of a (weak) direct sum of groups , where for any . As a result, the additive group of a graded algebra (considered as a module over the ring of integers) is a positively graded module. As an example of a graded algebra one can quote the algebra of polynomials over a field , where is the subspace generated by the monomials of degree ( ). One can also define a graded algebra in a more general manner as an algebra whose additive group can be represented as a direct sum of groups , where runs through a certain commutative semi-group and for any . The concept of a filtered algebra is closely connected with that of a graded algebra. It is in fact possible to define, in a natural manner, an ascending filtration on each graded algebra . Conversely, if is a filtered algebra , then one defines the graded algebra (where , ), which is called the graded algebra associated with . A graded ring is defined in a similar manner.
For an arbitrary group one may define a gradation of type on an algebra , i.e. where each is an additive subgroup of and for all . The group algebra over the field , as well as crossed products defined by a group morphism and a -cocycle , are examples of -graded algebras (see also Cross product). The use of -gradations that are not necessarily positive gradations allows one to consider the graded rings associated to -adic filtrations on a ring ; for an ideal of the -adic filtration is given by a descending chain , and then , where is negatively graded.